Bài 5: Phép Cộng Các Phân Thức Đại Số
Hướng dẫn Giải Bài 23 (Trang 46, SGK Toán 8, Tập 1)
<p>Làm các phép tính sau:</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mi>y</mi></mrow></mfrac></math>; b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac></math>; d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>.</mo></math></p>
<p><strong>Giải</strong></p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mi>y</mi><mrow><mi>x</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mi>y</mi><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mi>y</mi><mrow><mi>x</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>-</mo><mn>4</mn><mi>x</mi></mrow><mrow><mi>y</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mrow><mi>x</mi><mi>y</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mi>x</mi><mi>y</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mi>x</mi><mi>y</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac></math></p>
<p>= <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></math></p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo>+</mo><mi>x</mi><mo>-</mo><mn>14</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>6</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced></mrow><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math></p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>+</mo><mn>1</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>4</mn><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac></math></p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>4</mn><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac></math></p>
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